If you work with oscilloscopes
on a regular basis, you know know one of the first things you do (or should
do) is to calibrate the frequency response of the probe by hooking it onto the
squarewave port and tweaking the probe capacitor for no overshooting or undershooting
at the waveform edges, and then verify that the displayed amplitude is correct.
I remember being amazed during engineering courses at learning that any periodic
waveform can be described mathematically as the sum of sinewaves at various
frequencies, amplitudes, and phases. Knowing the theory behind those waveforms
- particularly standard ones like squarewaves, trianglewaves, sawtooths, etc.
- really helps in understanding what you see on the o-scope and in troubleshooting
problems. The same goes for interpreting the impulse and step function responses
as influenced by resistance, capacitance, and inductance effects. Perhaps the
most amazing thing I learned about squarewaves is that, based on the
phenomenon, anything short of an infinite series of additive sinewaves when
representing a squarewave results in an overshoot - albeit vanishingly minute
- at the edge. In the real world, complex reactive/resistive effects render
the effect undetectable.
July 1957 Radio & TV News
Wax nostalgic about and learn from the history of early
electronics. See articles from
Radio & Television News, published 1919-1959. All copyrights hereby
Practical Techniques of Square-Wave
By E. G. Louis
A square-wave generator and an oscilloscope are useful tools in designing
and servicing wide-band amplifiers. What troubles to look for with certain scope
By means of a Fourier analysis it is possible to show that an "ideal" square
wave, Fig. 1 (Curve 1) consists of a fundamental (Curve 2) sine wave whose
frequency is equal to that of the square wave, together with the 3rd (Curve
3), 5th, 7th, 9th, and higher odd harmonics, the amplitude of each decreasing
in direct proportion to its order. Theoretically, a perfect square wave consists
of a fundamental together with an infinite number of higher odd-harmonic signals.
Fig. 1. A square wave (1)
is made up of fundamental sine wave (2) and odd harmonics (3).
Fig. 2. Setup for square-wave
Fig. 3. Basic triode amplifier
If such a square-wave signal is applied to the input of an electrical circuit,
whether a filter network, amplifier, or other system, and the system does not
respond equally well to the fundamental and all higher harmonics, then the output
signal obtained will be distorted in a fashion indicative of the response characteristics
of the system under test. This is the basis of the square-wave test technique.
Since not only the fundamental, but all higher harmonics are applied simultaneously,
and an indication of the system's response to this wide range of signals is
obtained at once, square-wave testing provides an extremely rapid method for
checking such network characteristics as frequency response, phase shift, transient
response, etc. Because of the speed with which the square-wave test technique
can be applied and information obtained, this method becomes quite valuable
not only as an aid in the production testing and servicing of electronic equipment,
but can be applied with equal, if not greater, value to the requirements of
the practical design engineer.
In the past, the technique of square-wave testing has been confined largely
to testing high-fidelity audio amplifiers and wide-band amplifiers with a bandwidth
of perhaps several hundred kilocycles or 1 megacycle. With wide-band oscilloscopes
now commercially available at prices within the reach of even the moderate sized
experimental laboratory, and with square-wave generators on the market delivering
square waves with fast rise times to frequencies as high as 1 megacycle, this
valuable and easily applied technique can be applied to a very much greater
extent. It may be used for checking not only audio amplifiers, transformers,
and similar systems, but also for the check and design of wide-band scope and
radar amplifiers, video amplifiers, and similar wide-frequency-range networks.
Since the techniques of square-wave testing and analysis can be applied in
the same fashion irrespective of the end result in view, whether servicing,
design, or production test, we will try to simply outline the basic technique,
with the major emphasis on the application of the technique in the design and
service of wide-band amplifier circuits.
Fundamentally speaking, only two pieces of equipment are required to apply
the square-wave test technique, a square-wave generator and an oscilloscope.
The square-wave generator may consist of a sine-wave generator and a suitable
clipper amplifier where audio circuits and comparatively narrow-band circuits
are to be checked. Where the response of video amplifiers and similar wide-frequency-range
systems are to be checked, however, it is best to obtain a specially designed
In general, the square-wave generator should deliver perfect square waves
with a short rise time at frequencies from the lowest frequency response of
the system to be studied to a frequency one-tenth the highest frequency response
of the system. For practical laboratory work, a square-wave generator delivering
signals from 50 cps to about 500 kc. or 1 mc. with a rise time of at most .1
microsecond (and preferably less) will be found suitable. These signals may
be available over a continuously variable range, or only at four or five "spot"
values within this range. The output voltage should be easily varied from under
1 volt to at least 8 to 10 volts. Output impedance should be low, 600 ohms is
about the highest that can generally be tolerated, particularly at high frequencies.
The oscilloscope used must have characteristics that are superior to the
system under test. From a general viewpoint, its vertical amplifier should be
fiat below the lowest frequency square wave to be used in testing to a frequency
ten times higher than the highest frequency signal to be used (within 1 or 2
db). It should not, in itself, cause any appreciable tilt or overshoot to any
square-wave signal applied to its input within the range to be used for test
purposes. The vertical amplifier should have a sensitivity of at least .5 volt/inch
(peak-to-peak) and preferably more.
A linear time base should be available within the scope which permits observation
(with expanded sweep if necessary) of one cycle of both the highest and lowest
frequency square waves to be used in testing.
Applying the Technique
The basic set-up used for square-wave testing is illustrated in block diagram
form in Fig. 2. A square-wave signal is applied to the input of the system
to be tested, and the input and output signals observed on a cathode-ray oscilloscope.
Deviations from the original square-wave shape indicate certain characteristics
of the system under test.
Test leads, both to and from the equipment, should be as short as possible,
otherwise, with high-frequency signals, or signals with a short rise time, unnatural
peaking and overshoot may be introduced due to resonance in the connecting leads
The output signal should be observed a a point where the loading of the CRO
will not appreciably affect the circuit parameters. If a high-impedance, low-capacity
probe is used with the scope, then individual stage characteristics can be observed.
Limitations of Test Technique
Since a square wave contains only a fundamental and higher harmonics, it
is not ordinarily employed for checking the response of a system at frequencies
lower than its fundamental value. The exception to this is the case of a network
whose response is such that the fundamental of the square wave is changed in
some manner with respect to its higher frequency components. Such a condition
may cause a change in the square-wave shape indicative of the system's response
at lower frequencies.
Only odd harmonics of the square wave are present as part of the entire signal,
hence any sharp dips or holes in the response characteristics of the system
at specific frequencies falling between the odd harmonics may not show up in
a square-wave test. However, the response of most amplifiers varies in a smooth
manner and this limitation is minor.
In general, a square-wave test will not indicate distortion due to overload
or overdrive on an amplifier, unless the overload distortion varies with frequency.
The square wave is simply made more "square," and a sine-wave signal still must
be used for such tests.
Finally, since it is almost physically impossible to produce a "perfect"
square wave, and very difficult to detect changes in the square wave because
of deterioration of signals higher than the tenth harmonic, square-wave signals
should be available, and used, at approximately decade values. The exact number
of signals required for a complete test will depend on the bandwidth of the
system under test.
From a practical viewpoint, signals of 50 cps and 1 kc. are suitable for
testing usual amplifiers and transformers. For checking wide-band amplifiers
generally used in the lab, frequencies of 50 cps, 1 kc., 10 kc. and 100 kc.
may be used. For checking video amplifiers, signals of 50 cps, 1 kc., 10 kc.,
100 kc., and 500 kc. may be used. Signals as high as 1 mc. may be used for checking
special pulsing circuits.
Response to L. F. Square Waves
Typical patterns that may be obtained when a low-frequency square wave is applied
to an amplifier or network are shown in Fig. 4. A basic triode, resistance-coupled
amplifier age is shown in Fig. 3.
Fig. 4. Typical patterns
obtained with low-frequency square wave applied.
Fig. 5. Square-wave patterns
that may result from circuit deficiencies at the high frequencies with a good
Fig. 6. Illustrating rise
time measurement on a square waveform.
Fig. 7. Distributed capacities
and inductances in basic amplifier circuit.
Fig. 8. Series and shunt
peaking coils compensate for high-frequency losses.
Fig. 9. Basic amplifier response
curves showing effects of compensating circuits.
If the amplifier responds perfectly to the input square-wave signal, neither
attenuating nor accentuating the higher harmonics and causing no phase shift,
a perfect output square wave will be obtained which, except for amplitude, is
identical with the input signal, as shown in Fig. 4A.
A boost at the fundamental frequency of the square wave with respect to it
higher harmonics, but with no phase shift, will result in the rounded signal
shown in Fig. 4B. Conversely, a loss at the fundamental frequency will
result in a general dip in the square wave as shown in Fig. 4C, while a
dip in the response curve, causing a loss of a particular harmonic, will result
in a dip at one or more points in the square wave, as shown in Fig. 4D.
Leading phase shift at low frequencies, but without appreciable signal loss,
displaces the fundamental with respect to the harmonics, resulting in a tilted
square top as shown in Fig. 4E. This is generally due to too low time constant
in the RC coupling network (Cg1-Rg in Fig. 3). If
a loss of signal accompanies the phase shift, then the flat top will curve downward
as well as be slanted, as shown in Fig. 4F. An extreme case of too low
a time constant in the coupling network may cause differentiation of the signal,
allowing only the higher harmonics to pass and resulting in a peaked signal
as shown in Fig. 4G. Such a signal may also be obtained due to high-frequency
leakage around an attenuator circuit.
Where low-frequency compensation is added to the amplifier stage (Rc-Cc
in Fig. 3), overcompensation may result in the phase lagging at low frequencies,
causing the square wave to tilt in the opposite direction, as shown in Fig. 4H.
Irrespective of whether leading or lagging phase shift causes the square
wave flat top to tilt (Figs. 4E or 4H), the amount of tilt depends on the degree
of phase shift. A 10% slope will be obtained when the phase shift is 2° at the
fundamental frequency of the signal.
From a design viewpoint, conditions shown in Figs. 4E, 4F, or 4G generally
indicate (in Fig. 3) either that Cg1 or Rg, both,
should be increased in value, that Ck should be made larger, or that
insufficient low-frequency compensation has been added. This, in turn, means
that either Cc should be made smaller or Rc should be
made larger (with respect to RL). If the condition shown in Fig. 4H
is obtained, then the amount of low-frequency compensation should be lowered,
by either reducing the value of Rc or increasing the value of Cc.
From a servicing viewpoint, conditions indicated in Figs. 4E, 4F, or 4G generally
indicate (again in Fig. 3) either that Cg1 or Rg
has become lower in value (usually Cg1 will have partially opened
... fully open would result in Fig. 4G); that Ck has lost capacity,
that Cc has increased in capacity (unlikely) or that Rc
has dropped in value (which may happen due to overload). The condition of Fig. 4H
indicates, generally, that C. has lost capacity or developed high power factor.
It may also indicate that Rc has increased in value, but this is
not very likely.
The response of the amplifier to both low- and high-frequency signals must
be considered before a full analysis of circuit operation can be made. Both
tests, when taken together, give a much better picture of conditions in the
system under test.
Response to H. F. Square Waves
Circuit deficiencies at high frequencies may result in any of the patterns
shown in Fig. 5 (and even in some of those shown previously, particularly
as far as phase shift is concerned). A typical single-stage, RC-coupled triode
amplifier is shown in Fig. 7, together with some of the factors affecting
its response to high frequencies.
As in Fig. 4A, the "perfect" signal is shown in Fig. 5A. A loss
of higher frequency harmonic signals will result in a rounding of the leading
edge of the square wave as in Fig. 5B. The degree of rounding is dependent
on loss of high frequencies, and an extreme loss will result in the output square
wave approaching a sine wave in form. If almost all higher harmonics are lost,
the square wave may appear as in Fig. 5C.
Resonant circuits in the amplifier (or in the connecting leads) may cause
"ringing" and result in damped oscillations on the leading edge of the square
wave, as shown in Fig. 5D. The frequency at which the oscillations occur
can be determined approximately by multiplying the number of "cycles" that would
be present along the flat square top of the signal (if not damped) by twice
the fundamental frequency of the square wave. Where rapid damping occurs, or
where the resonant frequency is extremely high, only a small "overshoot," as
shown in Fig. 5E, may be obtained.
Since the rise time (time for square wave to go from 10% to 90% of its peak
value) is dependent on the number of higher harmonic signals present without
attenuation, this serves as a good indication of uniform frequency response
of an amplifier or network irrespective of whether rounding of the square wave
occurs or not.
An increase in rise time in a square wave is shown in Fig. 6.
A simple relationship, accurate enough for most practical design work, between
rise time and uniform frequency response of an amplifier, is as follows:
Maximum f (uniform response in mc.) = 1/2TR where TR
is the rise time in microseconds. (This relation holds true only where artificial
means, such as peaking coils, are not used to shorten the rise time.)
From a design viewpoint, conditions shown in Figs. 5B or 5C indicate that
the high-frequency response of an amplifier is not sufficient. This can be improved
by reducing the effect of distributed capacities shown in Fig. 7 by making
RL as small as is practicable for the gain desired, and then by using
series and shunt peaking coils, as shown in Fig. 8, to offset these capacities.
To obtain reasonable gain with a low RL, it may be necessary to go
to tubes having high mutual conductance. In such a case, the stage gain is equal
to the product of the load resistance and tube mutual conductance.
Distributed capacities are reduced by keeping leads short, parts and leads
above the chassis, and using miniature parts where economically feasible.
The inductance of the peaking coils is generally chosen so that resonance
will occur with the distributed capacities in the circuit at frequencies higher
than the highest frequency at which uniform response is desired in the amplifier.
In better amplifiers, these coils are usually made adjustable so that each unit
may be adjusted for best response for the individual distributed capacities
in that unit.
If these coils resonate at too low a frequency, or if insufficient damping
(RS1 and RS2 in Fig. 8) is used, a severe overshoot
may occur, or an oscillatory wave train may be set up, as shown in Fig. 5D.
Often a slight amount of overshoot (about 5% maximum) is desirable, as it
tends to shorten the rise time of the amplifier. Thus, the condition shown in
Fig. 5E would not always be considered objectionable.
A loss of higher frequency signals may occur if there is inductance in the
electrolytic capacitor used for cathode bypass (Lck in Fig. 7),
due to degeneration across Rk and the loss of gain at these frequencies.
In practical design, this can be offset by bypassing the electrolytic with a
small capacitor (around .005 μfd.), as at Ck1 in Fig. 8.
From a servicing viewpoint, conditions illustrated in Figs. 5B and 5C may
indicate that a peaking coil has become shorted or open (if shunted with a damping
resistor) or that .-RL has increased in value. In some cases it may
indicate an open in Ck1 (Fig. 8).
The condition shown in Fig. 5D may seldom be encountered, but generally
indicates that a damping resistor across a peaking coil has opened.
The condition of Fig. 5B, at higher frequencies, together with the condition
shown in Fig. 4H at lower frequencies, would indicate that Cc
(Fig. 3) has become open or dropped in capacity. Thus, it is practical
to use a combination of conditions to indicate a specific defect.
Square waves, observed with a cathode-ray oscilloscope, provide an efficient
and extremely rapid technique for testing and servicing systems designed to
pass a band of frequencies. The technique can also be used to good advantage
in design and production engineering, for adjusting circuits for proper operation,
and for determining optimum values of components.
The basic relationship between the frequency response of an amplifier or
network and its effect on square waves can be obtained by referring to Fig. 9.
A simple resistance-coupled amplifier may have the response shown in Curve
1. With this type of response curve, a low-frequency square wave will appear
as in Fig. 4E; at middle-range frequencies, as at Figs. 4A or 5A; while
a high-frequency square wave will appear as at Fig. 5B.
If low-frequency compensation is added, resulting in a boost at low frequencies
as shown in Curve 2, middle-range and high-frequency square waves will appear
as previously, but low-frequency square waves will generally be tilted in the
opposite direction as shown in Fig. 4H. With the proper amount of compensation,
a perfectly flat-top low-frequency square wave may be obtained, but this is
frequently difficult to maintain in production unless each unit is adjusted
When a peaking coil is used to provide a boost at high frequencies, the over-all
response may appear as shown in Curves 3 or 4, depending on the frequency of
peaking and the amount of damping. With a response as shown in Curve 3, a high-frequency
square wave may appear as in Fig. 5E, while middle and lower frequency
square waves will remain as previously. If the response is as in Curve 4, a
high-frequency square wave will appear as in Fig. 5D, and a middle-range
frequency square wave as in Fig. 5E.
If high-frequency compensation is obtained by reducing distributed capacities,
a high-frequency square wave can be made to approach the input signal in form,
as in Fig. 5A, and the response curve becomes as shown in Curve 5. Note
that the response falls off smoothly at higher frequencies.
The same result can be obtained by reducing RL, but this results
in reduced overall gain.
Posted February 5, 2014